Network Working Group | A. Biryukov |

Internet-Draft | D. Dinu |

Intended status: Informational | University of Luxembourg |

Expires: December 19, 2019 | D. Khovratovich |

ABDK Consulting | |

S. Josefsson | |

SJD AB | |

June 17, 2019 |

The memory-hard Argon2 password hash and proof-of-work function

draft-irtf-cfrg-argon2-05

This document describes the Argon2 memory-hard function for password hashing and proof-of-work applications. We provide an implementer-oriented description together with sample code and test vectors. The purpose is to simplify adoption of Argon2 for Internet protocols. This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF.

This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79.

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This Internet-Draft will expire on December 19, 2019.

Copyright (c) 2019 IETF Trust and the persons identified as the document authors. All rights reserved.

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- 1. Introduction
- 2. Notation and Conventions
- 3. Argon2 Algorithm
- 3.1. Argon2 Inputs and Outputs
- 3.2. Argon2 Operation
- 3.3. Variable-length hash function H'
- 3.4. Indexing
- 3.5. Compression function G
- 3.6. Permutation P
- 4. Parameter Choice
- 5. Example Code
- 6. Test Vectors
- 6.1. Argon2d Test Vectors
- 6.2. Argon2i Test Vectors
- 6.3. Argon2id Test Vectors
- 7. Acknowledgements
- 8. IANA Considerations
- 9. Security Considerations
- 9.1. Security as hash function and KDF
- 9.2. Security against time-space tradeoff attacks
- 9.3. Security for time-bounded defenders
- 9.4. Recommendations
- 10. References
- 10.1. Normative References
- 10.2. Informative References
- Authors' Addresses

This document describes the Argon2 memory-hard function for password hashing and proof-of-work applications. We provide an implementer oriented description together with sample code and test vectors. The purpose is to simplify adoption of Argon2 for Internet protocols. This document corresponds to version 1.3 of the Argon2 hash function.

Argon2 summarizes the state of the art in the design of memory-hard functions. It is a streamlined and simple design. It aims at the highest memory filling rate and effective use of multiple computing units, while still providing defense against tradeoff attacks. Argon2 is optimized for the x86 architecture and exploits the cache and memory organization of the recent Intel and AMD processors. Argon2 has one primary variant: Argon2id, and two supplementary variants: Argon2d and Argon2i. Argon2d uses data-dependent memory access, which makes it suitable for cryptocurrencies and proof-of-work applications with no threats from side-channel timing attacks. Argon2i uses data-independent memory access, which is preferred for password hashing and password-based key derivation. Argon2id works as Argon2i for the first half of the first iteration over the memory, and as Argon2d for the rest, thus providing both side-channel attack protection and brute-force cost savings due to time-memory tradeoffs. Argon2i makes more passes over the memory to protect from tradeoff attacks.

Argon2 can be viewed as a mode of operation over a fixed-input-length compression function G and a variable-input-length hash function H. Even though Argon2 can be potentially used with arbitrary function H, as long as it provides outputs up to 64 bytes, in this document it MUST be BLAKE2b.

For further background and discussion, see the Argon2 paper.

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in RFC 2119.

This document represents the consensus of the Crypto Forum Research Group (CFRG).

x^y --- integer x multiplied by itself integer y times

a*b --- multiplication of integer a and integer b

c-d --- substraction of integer c with integer d

E_f --- variable E with subscript index f

g / h --- integer g divided by integer h. The result is rational number

I(j) --- function I evaluated on integer parameter j

K || L --- string K concatenated with string L

a XOR b --- bitwise exclusive-or between bitstrings a and b

a mod b --- remainder of integer a modulo integer b, always in range [0, b-1]

a >>> n --- rotation of 64-bit string a to the right by n bits

trunc(a) --- the 64-bit value, truncated to the 32 least significant bits

floor(a) --- the largest integer not bigger than a

ceil(a) --- the smallest integer not smaller than a

extract(a, i) --- the i-th set of 32-bits from bitstring a, starting from 0-th

|A| --- the number of elements in set A

LE32(a) --- 32-bit integer a converted to bytestring in little endian. Example: 123456 (decimal) is 40 E2 01 00.

LE64(a) --- 64-bit integer a converted to bytestring in little endian. Example: 123456 (decimal) is 40 E2 01 00 00 00 00 00.

int32(s) --- 32-bit string s is converted to non-negative integer in little endian.

int64(s) --- 64-bit string s is converted to non-negative integer in little endian.

length(P) --- the bytelength of string P expressed as 32-bit integer

Argon2 has the following input parameters:

- Message string P, which is a password for password hashing applications. MUST have length from 0 to 2^(32) - 1 bytes.
- Nonce S, which is a salt for password hashing applications. MUST have length not greater than 2^(32)-1 bytes. 16 bytes is RECOMMENDED for password hashing. Salt SHOULD be unique for each password.
- Degree of parallelism p determines how many independent (but synchronizing) computational chains (lanes) can be run. It MUST be an integer value from 1 to 2^(24)-1.
- Tag length T MUST be an integer number of bytes from 4 to 2^(32)-1.
- Memory size m MUST be an integer number of kibibytes from 8*p to 2^(32)-1. The actual number of blocks is m', which is m rounded down to the nearest multiple of 4*p.
- Number of iterations t (used to tune the running time independently of the memory size) MUST be an integer number from 1 to 2^(32)-1.
- Version number v MUST be one byte 0x13.
- Secret value K is OPTIONAL. If used, it MUST have length not greater than 2^(32)-1 bytes.
- Associated data X is OPTIONAL. If used, it MUST have length not greater than 2^(32)-1 bytes.
- Type y of Argon2: MUST be 0 for Argon2d, 1 for Argon2i, 2 for Argon2id.

The Argon2 output, or "tag" is a string T bytes long.

Argon2 uses an internal compression function G with two 1024-byte inputs and a 1024-byte output, and an internal hash function H^x() with x being its output length in bytes. Here H^x() applied to string A is the BLAKE2b function, which takes (d,|dd|,kk=0,nn=x) as parameters where d is A padded to a multiple of 128 bytes and partitioned into 128-byte blocks. The compression function G is based on its internal permutation. A variable-length hash function H' built upon H is also used. G is described in Section Section 3.5 and H' is described in Section Section 3.3.

H_0 = H^(64)(LE32(p) || LE32(T) || LE32(m) || LE32(t) || LE32(v) || LE32(y) || LE32(length(P)) || P || LE32(length(S)) || S || LE32(length(K)) || K || LE32(length(X)) || X)

H_0 generation

m' = 4 * p * floor (m / 4p)

Memory allocation

B[i][0] = H'^(128)(H_0 || LE32(0) || LE32(i))

Lane starting blocks

B[i][1] = H'^(128)(H_0 || LE32(1) || LE32(i))

Second lane blocks

B[i][j] = G(B[i][j-1], B[l][z])

Further block generation

B[i][0] = G(B[i][q-1], B[l][z]) B[i][j] = G(B[i][j-1], B[l][z])

Further passes

C = B[0][q-1] XOR B[1][q-1] XOR ... XOR B[p-1][q-1]

Final block

The Argon2 operation is as follows.

- Establish H_0 as the 64-byte value as shown below.
- Allocate the memory as m' 1024-byte blocks where m' is derived as:
- Compute B[i][0] for all i ranging from (and including) 0 to (not including) p.
- Compute B[i][1] for all i ranging from (and including) 0 to (not including) p.
- Compute B[i][j] for all i ranging from (and including) 0 to (not including) p, and for all j ranging from (and including) 2) to (not including) q. The block indices l and z are determined for each i, j differently for Argon2d, Argon2i, and Argon2id (Section Section 3.4).
- If the number of iterations t is larger than 1, we repeat the steps however replacing the computations with the following expression:
- After t steps have been iterated, the final block C is computed as the XOR of the last column:
- The output tag is computed as H'^T(C).

if T <= 64 H'^T(A) = H^T(LE32(T)||A) else r = ceil(T/32)-2 V_1 = H^(64)(LE32(T)||A) V_2 = H^(64)(V_1) ... V_r = H^(64)(V_{r-1}) V_{r+1} = H^(T-32*r)(V_{r}) H'^T(X) = W_1 || W_2 || ... || W_r || V_{r+1}

Tag computation

Let V_i be a 64-byte block, and W_i be its first 32 bytes. Then we define:

To enable parallel block computation, we further partition the memory matrix into S = 4 vertical slices. The intersection of a slice and a lane is a segment of length q/S. Segments of the same slice can be computed in parallel and do not reference blocks from each other. All other blocks can be referenced.

slice 0 slice 1 slice 2 slice 3 ___/\___ ___/\___ ___/\___ ___/\___ / \ / \ / \ / \ +----------+----------+----------+----------+ | | | | | > lane 0 +----------+----------+----------+----------+ | | | | | > lane 1 +----------+----------+----------+----------+ | | | | | > lane 2 +----------+----------+----------+----------+ | ... ... ... | ... +----------+----------+----------+----------+ | | | | | > lane p - 1 +----------+----------+----------+----------+

Single-pass Argon2 with p lanes and 4 slices

J_1 = int32(extract(B[i][j-1], 1)) J_2 = int32(extract(B[i][j-1], 2))

Deriving J1,J2 in Argon2d

J_1 is given by the first 32 bits of block B[i][j-1], while J_2 is given by the next 32-bits of block B[i][j-1]:

( LE64(r) || LE64(l) || LE64(s) || LE64(m') || LE64(t) || LE64(y) || LE64(i) || ZERO ), where r -- the pass number l -- the lane number s -- the slice number m' -- the total number of memory blocks t -- the total number of passes y -- the Argon2 type (0 for Argon2d, 1 for Argon2i, 2 for Argon2id) i -- the counter (starts from 1 in each segment) ZERO -- the 968-byte zero string.

Input to compute J1,J2 in Argon2i

Each application of the 2-round compression function G in the counter mode gives 128 64-bit values X, which are viewed as X1||X2 and converted to J_1=int32(X1) and J_2=int32(X2). The first input to G is the all zero block and the second input to G is constructed as follows:

If the pass number is 0 and the slice number is 0 or 1, then compute J_1 and J_2 as for Argon2i, else compute J_1 and J_2 as for Argon2d.

The value of l = J_2 mod p gives the index of the lane from which the block will be taken. For the firt pass (r=0) and the first slice (s=0) the block is taken from the current lane.

The set W contains the indices that can be referenced according to the following rules:

- If l is the current lane, then W includes the indices of all blocks in the last S - 1 = 3 segments computed and finished, as well as the blocks computed in the current segment in the current pass excluding B[i][j-1].
- If l is not the current lane, then W includes the indices of all blocks in the last S - 1 = 3 segments computed and finished in lane l. If B[i][j] is the first block of a segment, then the very last index from W is excluded.

J_1 -> |W|(1 - J_1^2 / 2^(64))

Computing J1

We are going to take a block from W with a non-uniform distribution over [0, |W|) using the mapping

x = J_1^2 / 2^(32) y = (|W| * x) / 2^(32) z = |W| - 1 - y

Computing J1, part 2

To avoid floating point computation, the following approximation is used:

The value of z gives the reference block index in W.

P(A_0, A_1, ... ,A_7) = (B_0, B_1, ... ,B_7)

Blake round function P

Compression function G is built upon the BLAKE2b round function P. P operates on the 128-byte input, which can be viewed as 8 16-byte registers:

( Q_0, Q_1, Q_2, ... , Q_7) <- P( R_0, R_1, R_2, ... , R_7) ( Q_8, Q_9, Q_10, ... , Q_15) <- P( R_8, R_9, R_10, ... , R_15) ... (Q_56, Q_57, Q_58, ... , Q_63) <- P(R_56, R_57, R_58, ... , R_63) ( Z_0, Z_8, Z_16, ... , Z_56) <- P( Q_0, Q_8, Q_16, ... , Q_56) ( Z_1, Z_9, Z_17, ... , Z_57) <- P( Q_1, Q_9, Q_17, ... , Q_57) ... ( Z_7, Z_15, Z 23, ... , Z_63) <- P( Q_7, Q_15, Q_23, ... , Q_63)

Core of compression function G

Compression function G(X, Y) operates on two 1024-byte blocks X and Y. It first computes R = X XOR Y. Then R is viewed as a 8x8 matrix of 16-byte registers R_0, R_1, ... , R_63. Then P is first applied to each row, and then to each column to get Z:

G: (X, Y) -> R -> Q -> Z -> Z XOR R

+---+ +---+ | X | | Y | +---+ +---+ | | ---->XOR<---- --------| | \ / | +---+ | | R | | +---+ | | | \ / | P rowwise | | | \ / | +---+ | | Q | | +---+ | | | \ / | P columnwise | | | \ / | +---+ | | Z | | +---+ | | | \ / ------>XOR | \ /

Argon2 compression function G.

Finally, G outputs Z XOR R:

v_0 v_1 v_2 v_3 v_4 v_5 v_6 v_7 v_8 v_9 v_10 v_11 v_12 v_13 v_14 v_15

Matrix element labeling

GB(v_0, v_4, v_8, v_12) GB(v_1, v_5, v_9, v_13) GB(v_2, v_6, v_10, v_14) GB(v_3, v_7, v_11, v_15) GB(v_0, v_5, v_10, v_15) GB(v_1, v_6, v_11, v_12) GB(v_2, v_7, v_8, v_13) GB(v_3, v_4, v_9, v_14)

Feeding matrix elements to GB

a = (a + b + 2 * trunc(a) * trunc(b)) mod 2^(64) d = (d XOR a) >>> 32 c = (c + d + 2 * trunc(c) * trunc(d)) mod 2^(64) b = (b XOR c) >>> 24 a = (a + b + 2 * trunc(a) * trunc(b)) mod 2^(64) d = (d XOR a) >>> 16 c = (c + d + 2 * trunc(c) * trunc(d)) mod 2^(64) b = (b XOR c) >>> 63

Details of GB

Permutation P is based on the round function of BLAKE2b. The 8 16-byte inputs S_0, S_1, ... , S_7 are viewed as a 4x4 matrix of 64-bit words, where S_i = (v_{2*i+1} || v_{2*i}):

Argon2d is optimized for settings where the adversary does not get regular access to system memory or CPU, i.e. he can not run side-channel attacks based on the timing information, nor he can recover the password much faster using garbage collection. These settings are more typical for backend servers and cryptocurrency minings. For practice we suggest the following settings:

- Cryptocurrency mining, that takes 0.1 seconds on a 2 Ghz CPU using 1 core — Argon2d with 2 lanes and 250 MB of RAM.

Argon2id is optimized for more realistic settings, where the adversary possibly can access the same machine, use its CPU or mount cold-boot attacks. We suggest the following settings:

- Backend server authentication, that takes 0.5 seconds on a 2 GHz CPU using 4 cores — Argon2id with 8 lanes and 4 GiB of RAM.
- Key derivation for hard-drive encryption, that takes 3 seconds on a 2 GHz CPU using 2 cores - Argon2id with 4 lanes and 6 GiB of RAM.
- Frontend server authentication, that takes 0.5 seconds on a 2 GHz CPU using 2 cores - Argon2id with 4 lanes and 1 GiB of RAM.

We recommend the following procedure to select the type and the parameters for practical use of Argon2.

- Select the type y. If you do not know the difference between them or you consider side-channel attacks as viable threat, choose Argon2id.
- Figure out the maximum number h of threads that can be initiated by each call to Argon2.
- Figure out the maximum amount m of memory that each call can afford.
- Figure out the maximum amount x of time (in seconds) that each call can afford.
- Select the salt length. 128 bits is sufficient for all applications, but can be reduced to 64 bits in the case of space constraints.
- Select the tag length. 128 bits is sufficient for most applications, including key derivation. If longer keys are needed, select longer tags.
- If side-channel attacks are a viable threat, or if you're uncertain, enable the memory wiping option in the library call.
- Run the scheme of type y, memory m and h lanes and threads, using different number of passes t. Figure out the maximum t such that the running time does not exceed x. If it exceeds x even for t = 1, reduce m accordingly.
- Hash all the passwords with the just determined values m, h, and t.

void fill_block(const block *prev_block, const block *ref_block, block *next_block) { block blockR, block_tmp; unsigned i; copy_block(&blockR, ref_block); xor_block(&blockR, prev_block); copy_block(&block_tmp, &blockR); /* Now blockR = ref_block + prev_block and bloc_tmp = ref_block + prev_block */ /* Apply Blake2 on columns of 64-bit words: (0,1,...,15), then (16,17,..31)... finally (112,113,...127) */ for (i = 0; i < 8; ++i) { BLAKE2_ROUND_NOMSG( blockR.v[16 * i], blockR.v[16 * i + 1], blockR.v[16 * i + 2], blockR.v[16 * i + 3], blockR.v[16 * i + 4], blockR.v[16 * i + 5], blockR.v[16 * i + 6], blockR.v[16 * i + 7], blockR.v[16 * i + 8], blockR.v[16 * i + 9], blockR.v[16 * i + 10], blockR.v[16 * i + 11], blockR.v[16 * i + 12], blockR.v[16 * i + 13], blockR.v[16 * i + 14], blockR.v[16 * i + 15]); } /* Apply Blake2 on rows of 64-bit words: (0,1,16,17,...112,113), then (2,3,18,19,...,114,115), ... and finally (14,15,30,31,...,126,127) */ for (i = 0; i < 8; i++) { BLAKE2_ROUND_NOMSG( blockR.v[2 * i], blockR.v[2 * i + 1], blockR.v[2 * i + 16], blockR.v[2 * i + 17], blockR.v[2 * i + 32], blockR.v[2 * i + 33], blockR.v[2 * i + 48], blockR.v[2 * i + 49], blockR.v[2 * i + 64], blockR.v[2 * i + 65], blockR.v[2 * i + 80], blockR.v[2 * i + 81], blockR.v[2 * i + 96], blockR.v[2 * i + 97], blockR.v[2 * i + 112], blockR.v[2 * i + 113]); } copy_block(next_block, &block_tmp); xor_block(next_block, &blockR); }

Example code

void fill_block_with_xor(const block *prev_block, const block *ref_block, block *next_block) { block blockR, block_tmp; unsigned i; copy_block(&blockR, ref_block); xor_block(&blockR, prev_block); copy_block(&block_tmp, &blockR); /* Saving the next block contents for XOR over */ xor_block(&block_tmp, next_block); /* Now blockR = ref_block + prev_block and bloc_tmp = ref_block + prev_block + next_block*/ /* Apply Blake2 on columns of 64-bit words: (0,1,...,15) , then (16,17,..31),... and finally (112,113,...127) */ for (i = 0; i < 8; ++i) { BLAKE2_ROUND_NOMSG( blockR.v[16 * i], blockR.v[16 * i + 1], blockR.v[16 * i + 2], blockR.v[16 * i + 3], blockR.v[16 * i + 4], blockR.v[16 * i + 5], blockR.v[16 * i + 6], blockR.v[16 * i + 7], blockR.v[16 * i + 8], blockR.v[16 * i + 9], blockR.v[16 * i + 10], blockR.v[16 * i + 11], blockR.v[16 * i + 12], blockR.v[16 * i + 13], blockR.v[16 * i + 14], blockR.v[16 * i + 15]); } /* Apply Blake2 on rows of 64-bit words: (0,1,16,17,...112,113), then (2,3,18,19,...,114,115), ... and finally (14,15,30,31,...,126,127) */ for (i = 0; i < 8; i++) { BLAKE2_ROUND_NOMSG( blockR.v[2 * i], blockR.v[2 * i + 1], blockR.v[2 * i + 16], blockR.v[2 * i + 17], blockR.v[2 * i + 32], blockR.v[2 * i + 33], blockR.v[2 * i + 48], blockR.v[2 * i + 49], blockR.v[2 * i + 64], blockR.v[2 * i + 65], blockR.v[2 * i + 80], blockR.v[2 * i + 81], blockR.v[2 * i + 96], blockR.v[2 * i + 97], blockR.v[2 * i + 112], blockR.v[2 * i + 113]); } copy_block(next_block, &block_tmp); xor_block(next_block, &blockR); }

Example code page 2

void generate_addresses(const argon2_instance_t *instance, const argon2_position_t *position, uint64_t *pseudo_rands) { block zero_block, input_block, address_block,tmp_block; uint32_t i; init_block_value(&zero_block, 0); init_block_value(&input_block, 0); if (instance != NULL && position != NULL) { input_block.v[0] = position->pass; input_block.v[1] = position->lane; input_block.v[2] = position->slice; input_block.v[3] = instance->memory_blocks; input_block.v[4] = instance->passes; input_block.v[5] = instance->type; for (i = 0; i < instance->segment_length; ++i) { if (i % ARGON2_ADDRESSES_IN_BLOCK == 0) { input_block.v[6]++; init_block_value(&tmp_block, 0); init_block_value(&address_block, 0); fill_block_with_xor(&zero_block, &input_block, &tmp_block); fill_block_with_xor(&zero_block, &tmp_block, &address_block); } pseudo_rands[i] = address_block.v[i % ARGON2_ADDRESSES_IN_BLOCK]; } }

Example code page 3

void fill_segment(const argon2_instance_t *instance, argon2_position_t position) { block *ref_block = NULL, *curr_block = NULL; uint64_t pseudo_rand, ref_index, ref_lane; uint32_t prev_offset, curr_offset; uint32_t starting_index; uint32_t i; int data_independent_addressing; /* Pseudo-random values that determine the reference block position */ uint64_t *pseudo_rands = NULL; if (instance == NULL) { return; } data_independent_addressing = (instance->type == Argon2_i); pseudo_rands = (uint64_t *)malloc(sizeof(uint64_t) * (instance->segment_length)); if (pseudo_rands == NULL) { return; } if (data_independent_addressing) { generate_addresses(instance, &position, pseudo_rands); } starting_index = 0; if ((0 == position.pass) && (0 == position.slice)) { /* we have already generated the first two blocks */ starting_index = 2; } /* Offset of the current block */ curr_offset = position.lane * instance->lane_length + position.slice * instance->segment_length + starting_index; if (0 == curr_offset % instance->lane_length) { /* Last block in this lane */ prev_offset = curr_offset + instance->lane_length - 1; } else { /* Previous block */ prev_offset = curr_offset - 1; } for (i = starting_index; i < instance->segment_length; ++i, ++curr_offset, ++prev_offset) { /*1.1 Rotating prev_offset if needed */ if (curr_offset % instance->lane_length == 1) { prev_offset = curr_offset - 1; } /* 1.2 Computing the index of the reference block */ /* 1.2.1 Taking pseudo-random value from the previous block */ if (data_independent_addressing) { pseudo_rand = pseudo_rands[i]; } else { pseudo_rand = instance->memory[prev_offset].v[0]; } /* 1.2.2 Computing the lane of the reference block */ ref_lane = ((pseudo_rand >> 32)) % instance->lanes; if ((position.pass == 0) && (position.slice == 0)) { /* Can not reference other lanes yet */ ref_lane = position.lane; } /* 1.2.3 Computing the number of possible reference block within the lane. */ position.index = i; ref_index = index_alpha(instance, &position, pseudo_rand & 0xFFFFFFFF, ref_lane == position.lane); /* 2 Creating a new block */ ref_block = instance->memory + instance->lane_length * ref_lane + ref_index; curr_block = instance->memory + curr_offset; if (instance->version == ARGON2_OLD_VERSION_NUMBER) { /* version 1.2.1 and earlier: overwrite, not XOR */ fill_block(instance->memory + prev_offset, ref_block, curr_block); } else { if(0 == position.pass) { fill_block(instance->memory + prev_offset, ref_block, curr_block); } else { fill_block_with_xor(instance->memory + prev_offset, ref_block, curr_block); } } } free(pseudo_rands); }

Example code page 4

uint32_t index_alpha(const argon2_instance_t *instance, const argon2_position_t *position, uint32_t pseudo_rand, int same_lane) { /* * Pass 0: * This lane : all already finished segments plus already * constructed blocks in this segment * Other lanes : all already finished segments * Pass 1+: * This lane : (SYNC_POINTS - 1) last segments plus * already constructed blocks in this segment * Other lanes : (SYNC_POINTS - 1) last segments */ uint32_t reference_area_size; uint64_t relative_position; uint32_t start_position, absolute_position; if (0 == position->pass) { /* First pass */ if (0 == position->slice) { /* First slice */ reference_area_size = position->index - 1; /* all but the previous */ } else { if (same_lane) { /* The same lane => add current segment */ reference_area_size = position->slice * instance->segment_length + position->index - 1; } else { reference_area_size = position->slice * instance->segment_length + ((position->index == 0) ? (-1) : 0); } } } else { /* Second pass */ if (same_lane) { reference_area_size = instance->lane_length - instance->segment_length + position->index - 1; } else { reference_area_size = instance->lane_length - instance->segment_length + ((position->index == 0) ? (-1) : 0); } } /* 1.2.4. Mapping pseudo_rand to 0..<reference_area_size-1> and produce relative position */ relative_position = pseudo_rand; relative_position = relative_position * relative_position >> 32; relative_position = reference_area_size - 1 - (reference_area_size * relative_position >> 32); /* 1.2.5 Computing starting position */ start_position = 0; if (0 != position->pass) { start_position = (position->slice == ARGON2_SYNC_POINTS - 1) ? 0 : (position->slice + 1) * instance->segment_length; } /* 1.2.6. Computing absolute position */ absolute_position = (start_position + relative_position) % instance->lane_length; /* absolute position */ return absolute_position; }

Example code page 5

int fill_memory_blocks(argon2_instance_t *instance) { uint32_t r, s; argon2_thread_handle_t *thread = NULL; argon2_thread_data *thr_data = NULL; if (instance == NULL || instance->lanes == 0) { return ARGON2_THREAD_FAIL; } /* 1. Allocating space for threads */ thread = calloc(instance->lanes, sizeof(argon2_thread_handle_t)); if (thread == NULL) { return ARGON2_MEMORY_ALLOCATION_ERROR; } thr_data = calloc(instance->lanes, sizeof(argon2_thread_data)); if (thr_data == NULL) { free(thread); return ARGON2_MEMORY_ALLOCATION_ERROR; } for (r = 0; r < instance->passes; ++r) { for (s = 0; s < ARGON2_SYNC_POINTS; ++s) { int rc; uint32_t l; /* 2. Calling threads */ for (l = 0; l < instance->lanes; ++l) { argon2_position_t position; /* 2.1 Join a thread if limit is exceeded */ if (l >= instance->threads) { rc = argon2_thread_join(thread[l - instance->threads]); if (rc) { free(thr_data); free(thread); return ARGON2_THREAD_FAIL; } } /* 2.2 Create thread */ position.pass = r; position.lane = l; position.slice = (uint8_t)s; position.index = 0; /* preparing the thread input */ thr_data[l].instance_ptr = instance; memcpy(&(thr_data[l].pos), &position, sizeof(argon2_position_t)); rc = argon2_thread_create(&thread[l], &fill_segment_thr, (void *)&thr_data[l]); if (rc) { free(thr_data); free(thread); return ARGON2_THREAD_FAIL; } /* fill_segment(instance, position); */ /*Non-thread equivalent of the lines above */ } /* 3. Joining remaining threads */ for (l = instance->lanes - instance->threads; l < instance->lanes; ++l) { rc = argon2_thread_join(thread[l]); if (rc) { return ARGON2_THREAD_FAIL; } } } } if (thread != NULL) { free(thread); } if (thr_data != NULL) { free(thr_data); } return ARGON2_OK; }

Example code page 6

This section contains test vectors for Argon2.

======================================= Argon2d version number 19 ======================================= Memory: 32 KiB Iterations: 3 Parallelism: 4 lanes Tag length: 32 bytes Password[32]: 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 Salt[16]: 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 Secret[8]: 03 03 03 03 03 03 03 03 Associated data[12]: 04 04 04 04 04 04 04 04 04 04 04 04 Pre-hashing digest: b8 81 97 91 a0 35 96 60 bb 77 09 c8 5f a4 8f 04 d5 d8 2c 05 c5 f2 15 cc db 88 54 91 71 7c f7 57 08 2c 28 b9 51 be 38 14 10 b5 fc 2e b7 27 40 33 b9 fd c7 ae 67 2b ca ac 5d 17 90 97 a4 af 31 09 After pass 0: Block 0000 [ 0]: db2fea6b2c6f5c8a Block 0000 [ 1]: 719413be00f82634 Block 0000 [ 2]: a1e3f6dd42aa25cc Block 0000 [ 3]: 3ea8efd4d55ac0d1 ... Block 0031 [124]: 28d17914aea9734c Block 0031 [125]: 6a4622176522e398 Block 0031 [126]: 951aa08aeecb2c05 Block 0031 [127]: 6a6c49d2cb75d5b6 After pass 1: Block 0000 [ 0]: d3801200410f8c0d Block 0000 [ 1]: 0bf9e8a6e442ba6d Block 0000 [ 2]: e2ca92fe9c541fcc Block 0000 [ 3]: 6269fe6db177a388 ... Block 0031 [124]: 9eacfcfbdb3ce0fc Block 0031 [125]: 07dedaeb0aee71ac Block 0031 [126]: 074435fad91548f4 Block 0031 [127]: 2dbfff23f31b5883 After pass 2: Block 0000 [ 0]: 5f047b575c5ff4d2 Block 0000 [ 1]: f06985dbf11c91a8 Block 0000 [ 2]: 89efb2759f9a8964 Block 0000 [ 3]: 7486a73f62f9b142 ... Block 0031 [124]: 57cfb9d20479da49 Block 0031 [125]: 4099654bc6607f69 Block 0031 [126]: f142a1126075a5c8 Block 0031 [127]: c341b3ca45c10da5 Tag: 51 2b 39 1b 6f 11 62 97 53 71 d3 09 19 73 42 94 f8 68 e3 be 39 84 f3 c1 a1 3a 4d b9 fa be 4a cb

======================================= Argon2i version number 19 ======================================= Memory: 32 KiB Iterations: 3 Parallelism: 4 lanes Tag length: 32 bytes Password[32]: 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 Salt[16]: 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 Secret[8]: 03 03 03 03 03 03 03 03 Associated data[12]: 04 04 04 04 04 04 04 04 04 04 04 04 Pre-hashing digest: c4 60 65 81 52 76 a0 b3 e7 31 73 1c 90 2f 1f d8 0c f7 76 90 7f bb 7b 6a 5c a7 2e 7b 56 01 1f ee ca 44 6c 86 dd 75 b9 46 9a 5e 68 79 de c4 b7 2d 08 63 fb 93 9b 98 2e 5f 39 7c c7 d1 64 fd da a9 After pass 0: Block 0000 [ 0]: f8f9e84545db08f6 Block 0000 [ 1]: 9b073a5c87aa2d97 Block 0000 [ 2]: d1e868d75ca8d8e4 Block 0000 [ 3]: 349634174e1aebcc ... Block 0031 [124]: 975f596583745e30 Block 0031 [125]: e349bdd7edeb3092 Block 0031 [126]: b751a689b7a83659 Block 0031 [127]: c570f2ab2a86cf00 After pass 1: Block 0000 [ 0]: b2e4ddfcf76dc85a Block 0000 [ 1]: 4ffd0626c89a2327 Block 0000 [ 2]: 4af1440fff212980 Block 0000 [ 3]: 1e77299c7408505b ... Block 0031 [124]: e4274fd675d1e1d6 Block 0031 [125]: 903fffb7c4a14c98 Block 0031 [126]: 7e5db55def471966 Block 0031 [127]: 421b3c6e9555b79d After pass 2: Block 0000 [ 0]: af2a8bd8482c2f11 Block 0000 [ 1]: 785442294fa55e6d Block 0000 [ 2]: 9256a768529a7f96 Block 0000 [ 3]: 25a1c1f5bb953766 ... Block 0031 [124]: 68cf72fccc7112b9 Block 0031 [125]: 91e8c6f8bb0ad70d Block 0031 [126]: 4f59c8bd65cbb765 Block 0031 [127]: 71e436f035f30ed0 Tag: c8 14 d9 d1 dc 7f 37 aa 13 f0 d7 7f 24 94 bd a1 c8 de 6b 01 6d d3 88 d2 99 52 a4 c4 67 2b 6c e8

======================================= Argon2id version number 19 ======================================= Memory: 32 KiB, Iterations: 3, Parallelism: 4 lanes, Tag length: 32 bytes Password[32]: 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 Salt[16]: 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 Secret[8]: 03 03 03 03 03 03 03 03 Associated data[12]: 04 04 04 04 04 04 04 04 04 04 04 04 Pre-hashing digest: 28 89 de 48 7e b4 2a e5 00 c0 00 7e d9 25 2f 10 69 ea de c4 0d 57 65 b4 85 de 6d c2 43 7a 67 b8 54 6a 2f 0a cc 1a 08 82 db 8f cf 74 71 4b 47 2e 94 df 42 1a 5d a1 11 2f fa 11 43 43 70 a1 e9 97 After pass 0: Block 0000 [ 0]: 6b2e09f10671bd43 Block 0000 [ 1]: f69f5c27918a21be Block 0000 [ 2]: dea7810ea41290e1 Block 0000 [ 3]: 6787f7171870f893 ... Block 0031 [124]: 377fa81666dc7f2b Block 0031 [125]: 50e586398a9c39c8 Block 0031 [126]: 6f732732a550924a Block 0031 [127]: 81f88b28683ea8e5 After pass 1: Block 0000 [ 0]: 3653ec9d01583df9 Block 0000 [ 1]: 69ef53a72d1e1fd3 Block 0000 [ 2]: 35635631744ab54f Block 0000 [ 3]: 599512e96a37ab6e ... Block 0031 [124]: 4d4b435cea35caa6 Block 0031 [125]: c582210d99ad1359 Block 0031 [126]: d087971b36fd6d77 Block 0031 [127]: a55222a93754c692 After pass 2: Block 0000 [ 0]: 942363968ce597a4 Block 0000 [ 1]: a22448c0bdad5760 Block 0000 [ 2]: a5f80662b6fa8748 Block 0000 [ 3]: a0f9b9ce392f719f ... Block 0031 [124]: d723359b485f509b Block 0031 [125]: cb78824f42375111 Block 0031 [126]: 35bc8cc6e83b1875 Block 0031 [127]: 0b012846a40f346a Tag: 0d 64 0d f5 8d 78 76 6c 08 c0 37 a3 4a 8b 53 c9 d0 1e f0 45 2d 75 b6 5e b5 25 20 e9 6b 01 e6 59

We thank greatly the following authors who helped a lot in preparing and reviewing this document: Jean-Philippe Aumasson, Samuel Neves, Joel Alwen, Jeremiah Blocki, Bill Cox, Arnold Reinhold, Solar Designer, Russ Housley, Stanislav Smyshlyaev, Kenny Paterson, Alexey Melnikov.

None.

The collision and preimage resistance levels of Argon2 are equivalent to those of the underlying BLAKE2b hash function. To produce a collision, 2^(256) inputs are needed. To find a preimage, 2^(512) inputs must be tried.

The KDF security is determined by the key length and the size of the internal state of hash function H'. To distinguish the output of keyed Argon2 from random, minimum of (2^(128),2^length(K)) calls to BLAKE2b are needed.

Time-space tradeoffs allow computing a memory-hard function storing fewer memory blocks at the cost of more calls to the internal comression function. The advantage of tradeoff attacks is measured in the reduction factor to the time-area product, where memory and extra compression function cores contribute to the area, and time is increased to accomodate the recomputation of missed blocks. A high reduction factor may potentially speed up preimage search.

The best attacks on the 1-pass and 2-pass Argon2i is the low-storage attack described in [CBS16], which reduces the time-area product (using the peak memory value) by the factor of 5. The best attack on 3-pass and more Argon2i is [AB16] with reduction factor being a function of memory size and the number of passes. For 1 gibibyte of memory: 3 for 3 passes, 2.5 for 4 passes, 2 for 6 passes. The reduction factor grows by about 0.5 with every doubling the memory size. To completely prevent time-space tradeoffs from [AB16], the number of passes MUST exceed binary logarithm of memory minus 26. Asymptotically, the best attack on 1-pass Argon2i is given in [BZ17] with maximal advantage of the adversary upper bounded by O(m^(0.233)) where m is the number of blocks. This attack is also asymptotically optimal as [BZ17] also prove the upper bound on any attack of O(m^(0.25)).

The best tradeoff attack on t-pass Argon2d is the ranking tradeoff attack, which reduces the time-area product by the factor of 1.33.

The best attack on Argon2id can be obtained by complementing the best attack on the 1-pass Argon2i with the best attack on a multi-pass Argon2d. Thus the best tradeoff attack on 1-pass Argon2id is the combined low-storage attack (for the first half of the memory) and the ranking attack (for the second half), which bring together the factor of about 2.1. The best tradeoff attack on t-pass Argon2id is the ranking tradeoff attack, which reduces the time-area product by the factor of 1.33.

A bottleneck in a system employing the password-hashing function is often the function latency rather than memory costs. A rational defender would then maximize the bruteforce costs for the attacker equipped with a list of hashes, salts, and timing information, for fixed computing time on the defender’s machine. The attack cost estimates from [AB16] imply that for Argon2i, 3 passes is almost optimal for the most of reasonable memory sizes, and that for Argon2d and Argon2id, 1 pass maximizes the attack costs for the constant defender time.

The Argon2id variant with t=1 and maximum available memory is recommended as a default setting for all environments. This setting is secure against side-channel attacks and maximizes adversarial costs on dedicated bruteforce hardware.

[RFC2119] |
Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", RFC 2119, March 1997. |

[RFC7693] |
Saarinen, M-J. and J-P. Aumasson, "The BLAKE2 Cryptographic Hash and Message Authentication Code (MAC)", RFC 7693, DOI 10.17487/RFC7693, November 2015. |